Much of the reform in mathematics education advocates
teachers supporting appropriate use of technology and teacher knowledge
structures that incorporate knowledge about subject matter, learners,
pedagogy, curriculum, and schools (Ball & Stacey, 2005; National
Council of Teachers of Mathematics [NCTM], 2000; Niess, 2005). Integration
of such techniques in effective manners allows teachers and students to
access advanced concepts, incite mathematical discourse, and represent
abstract concepts. Mishra and Koehler’s (2006) technological pedagogical
content knowledge (TPCK) framework lies at the core of understanding how
technology can help remedy some of the problems of teaching and learning.
TPCK brings together knowledge about content, pedagogy, and technology as
interconnecting factors affecting the development of effective teaching
with technology (Mishra & Koehler, 2006).
Mathematical TPCK, dissimilar from knowledge of all
three concepts individually, refers to the intersection of knowledge of
mathematics with knowledge of technology and with knowledge of teaching and
learning. It underlines the notion of using technology as a tool to
construct teacher and student mathematical knowledge (Mishra & Koehler,
2006). In order for technology to become a tool for learning mathematics,
mathematics teachers must develop “an overarching conception of their
subject matter with respect to technology and what it means to teach with
technology – technology pedagogical content knowledge (TPCK)” (Niess,
2005). Shulman (1987) defined content knowledge as the knowledge about the
subject (i.e., knowledge of mathematics and mathematical representations),
while knowledge of students, knowledge of teaching, and knowledge of
educational contexts characterize pedagogical content knowledge. The sum
and intersection of technological knowledge, pedagogical knowledge, and
content knowledge serve as a framework for effective mathematics teaching
and learning.
To be prepared to teach mathematics adequately, teachers
must have a comprehensive understanding of TPCK. As mathematics teachers
think about teaching with technology, they should concurrently consider how
to teach mathematical concepts in such a way that students can experiment
with ideas, make conjectures, test hypotheses, and form generalizations.
Likewise, in order for technology to support effective
mathematics teaching, teachers must develop or use appropriate mathematics
tasks that capitalize on the strengths of technology. Because of the wealth
of available technology resources, a challenge for teachers is in learning
how to selective appropriate and effective technology tools for mathematics
teaching and learning. As views of technology as a demonstration tool
expand to views of technology as more of a knowledge construction tool,
teachers will experience how to select, evaluate, design, teach, and learn
using this powerful resource.
As the NCTM (2000) highlighted, technology integration
in the teaching and learning of mathematics is a necessity. As the
necessity and availability of technology in mathematics classrooms
increases, so must supporting teachers in their practices, professional
development, and development of TPCK (Newby, Stepich, Lehman, &
Russell, 2000; Roblyer & Edwards, 2000). Although preparing teachers to
use technology appropriately is a complex task (Mergendoller, 1994),
research suggests that such professional development must entail both
conceptual and pedagogical issues. An important challenge is to
identify how to prepare mathematics teachers to teach in the 21st century
using TPCK and what they need to know to be able to do so.
This initiative provided professional development to 20
middle school teachers from six different schools in the same area.
Although the schools were in the same district, three of the schools were
in a rural setting and three in an urban setting. Participating teachers,
all women from various backgrounds, applied to participate in the project,
were paid a small stipend for attending sessions, and were given various
mathematics materials and supplies (e.g., TI Nspire technology, algebra
tiles classroom sets, and algebra blocks) to supplement their
teaching.
Only two of the teachers had more than 2 years of
Algebra I teaching experience. Slightly over two fifths of the teachers had
been teaching for at least 10 years, three of the teachers had 3 or fewer
years of teaching experience, and the remaining teachers had teaching
experience ranging from 3 to 10 years.
With the exception of one multigrade charter school
teacher, all participants were public school teachers teaching at least one
eighth-grade Algebra I course, with an average of 21 students in each
class. All 20 participating teachers had middle school general
teaching certifications; none had secondary mathematics certification.
The district, rated “Academically Acceptable” by the
state education agency for the past 2 years, had a 60% pass rate on the
state standardized eighth-grade mathematics assessment exam. Over 80% of
the district’s student population was classified as minorities, with 16% of
the students classified as English language learners. Eighty-two percent of
the district’s students were eligible for free or reduced price lunch
programs.
In response to a district mandate for all eighth-grade
students to be enrolled in Algebra I, all participating teachers began
teaching eighth-grade Algebra I for the first time during the academic year
following the beginning of the professional development sessions. Each
teacher received 120 professional development hours, including 60 hours of
summer professional development and 60 hours of academic year professional
development, each consisting of fifteen 4-hour sessions. Summer sessions
focused on conceptual knowledge, while the academic year agenda targeted
pedagogical techniques for developing and implementing effective Algebra I
classroom activities and instruction for all students, particularly those
from underrepresented groups.
The primary goals of the professional development
sessions were to (a) offer participating teachers new opportunities to
creatively formulate and communicate TPCK, (b) increase their understanding
of algebraic concepts, and (c) develop their problem-solving skills with an
emphasis on modeling concepts and using technology. The professional
development program guided teachers in learning about planning for teaching
and learning algebra using technology. Teachers increased their level of
content knowledge, learned to develop student-centered activities,
identified specific activities that guided their planning and teaching of
algebra, increased their knowledge and appropriate use of technology in the
teaching and learning of algebra, explored eighth-grade algebra projects,
examined how algebra fits into the curriculum, experimented with
instructional strategies, and engaged in constant mathematics dialog.
Additional instructional activities included writing in electronic
journals, designing technology-based curricular materials, using graphing
calculators, and using mathematical software.
As detailed in an outline of the summer and academic
year sessions in Appendix A and B, respectively, sessions encouraged
cooperative work, reflection on teaching experiences, and exploration of a
variety of solution strategies. The sessions were organized and conducted
by the author, a mathematics educator charged with the responsibility of
preparing preservice mathematics teachers and delivering continuing
education to in-service mathematics teachers to advance TPCK and a
colleague with numerous years of experience teaching mathematics at the
postsecondary level. A discussion of three session experiences is presented
later in the article.
Participant journal entries and observations of
interactions and discussions between participants were categorized and
analyzed using a TPCK content analysis framework (Mishra & Koehler,
2006). Identified participant discussions, interactions, and perceived
benefits of participating in the project were noted in one or more of four
categories developed to describe general themes in varying levels of TPCK
development. The four categories included technological pedagogical
knowledge (TPK), technological content knowledge (TCK), pedagogical content
knowledge (PCK), and technological pedagogical content knowledge (TPCK).
Descriptors or themes reflecting participants’
development of an understanding of how technology usage serves as an avenue
for multiple representations of algebraic concepts were sorted into the TCK
category, and those indicative of participants’ understanding of how
teaching algebra changes as a result of using a variety of available
technologies were grouped in the TPK category. The PCK category highlighted
indicators of how participants’ understanding of algebraic concepts
influenced their teaching of algebra.
Naturally, the TPCK category grouped blends of all three
categories. Responses in one category often intersected with one of the
three other content analysis categorizations. All observations and journal
entry details were categorized using at least one of the four descriptors
to describe participants’ levels of TPCK in teaching and learning algebra.
Conclusions
This paper presented components of a professional
development project that offered new experiences in creative work for
in-service Algebra I teachers. A primary goal of this professional
development project was to develop and advance teachers’ TPCK. Analysis of
the data under the TPCK framework was useful in diagnosing in-service
mathematics teachers’ need to intersect, rather than isolate, these three
knowledge bases. Participating teachers were not labeled as a content
expert, technology expert, or pedagogical expert but instead classified as
developing professionalism simultaneously in all three components.
Analysis of the professional development sessions revealed
the need for more professional development focused on enhancing teachers’
ability to connect mathematical ideas using technology and on their
pedagogical and content skills to work with multiple representations of
mathematical ideas. This trend was noted in instances of data analysis,
where teachers focused more on developing TCK in comparison to TPCK or PCK.
Another important finding from the data analysis is that teachers need
opportunities to explore how to integrate nontraditional forms of technology
effectively into both routine and nonroutine algebraic classroom
instruction. More specifically, teachers need help in making the transition
from using technology-based manipulatives for illustrating mathematical
concepts (TCK) to utilizing these tools as means for exploration and
discovery leading to students’ deep conceptual understanding of mathematics
(TPCK).
In an informal exit discussion, all 20 participating
teachers answered yes to the question “Did the professional development
sessions provide you with valuable insights of how to use technology to
explore, investigate, and verify new mathematical situations?” Responses to
this question showed clear evidence of the teachers’ TPCK advancement.
Challenges encountered in the professional development sessions included
getting teachers to focus on technology, issues, and pedagogy collectively
in lieu of focusing simply on technology and the teachers’ disposition
toward new technology. In addition, no emphasis was placed on considering
possible limitations of integrating technology in the teaching and learning
of algebra. The sessions primarily focused on identifying the benefits of
developing, exploring, and advancing the teachers’ TPCK.
|
